Representation of $n$-th derivative as the sum

Question

Not so long time working with derivatives I have noticed, that if $f(x)=a^{b^x}$, so for $n\geq0$ $$f^{(n)}(x)=(\log b) f(x) \sum\limits_{k=0}^{n} {n \brace k} (\log f(x))^k$$ and also if $g(x)=a^{\frac{1}{x}}$, so for $n>0$ $$g^{(n)}(x)=(-\frac{1}{x})^n g(x) \sum\limits_{k=1}^{n} \frac{n!}{k!} \binom{n-1}{k-1}(\log g(x))^k$$ How can we prove it? Is there a general method for finding representation of $n$-th derivative as the sum?